Recently, I'm tired of those theoretical parts on commuative algebra. So I hope that someone could recommend me some good textbooks on SINGULAR and Macaulay 2. And I'm wondering whether SINGULAR is better that Macaulay 2?

Your question does not have an answer, for 'good' is not an absolute notion. Computer algebra systems are good or bad for specific purposes, and the best program in the universe for me might very well be completely useless for you! Maybe you could tell us want you want to do with SINGULAR or Macaulay, and someone knowledgeable with them can tell you which is the best fit.
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Mariano Suárez-Alvarez♦May 3 '10 at 3:56

5 Answers
5

Macauly 2 and Singular share the same computational engine (singular) so none if them is "better" in any real sense

the best book+software combination I know of is COCOA plus the two volumes of "computational commutative algebra". My "issue" with the singular book is that it's too basic, and with the Macauly book that it's simply a compendium of articles, and not a real text book.

Well, not all programming languages are equivalent in any meaningful sense, despite most of them being computationally equivalent!
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Mariano Suárez-Alvarez♦May 3 '10 at 5:41

In my opinion, the usefulness of Singular program is very limited. For example, can we define a polynomial ring over a general noetherian ring(e.g. the integral ring Z)?
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TmobiusXMay 3 '10 at 7:18

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@Mariano, point taken: Singular is indeed (more) aimed at people who want to write extensions, and Macauly at people who simply want to compute something. @Tmobius: whats wrong with "ring r 0,(x),ls" ?
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David LehaviMay 3 '10 at 7:42

@David,Lehavi, what does "ring r0,(x)" mean? Could you explain it explicitly?
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TmobiusXMay 4 '10 at 1:39

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As Bart Snapp points out in an answer, it is not true that Macaulay2 and Singular share the same computational engine. "Macaulay2's engine code for polynomials, Groebner bases, and free resolutions is its own, written by Mike Stillman. What is true is that Singular and Macaulay2 both use two libraries written by the Singular group: Singular-Factory and Singular-Libfac. Macaulay2 uses those libraries for factoring of polynomials, gcd of polynomials, characteristic series (which is the core of the algorithm for computing minimal primes)." [quoted from groups.google.com/group/macaulay2 ]
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Graham LeuschkeMay 24 '10 at 18:51

In addition to the Macaulay 2 book, there's a book called "A Singular Introduction to Commutative Algebra" by Greuel and Pfister (developers of Singular). Both of these books are good for references, but the books by Cox, Little, and O'Shea are better for reading, I think.